A091485 Number of labeled 2,3 cacti (triangular cacti with bridges).

1, 1, 4, 28, 290, 3996, 68992, 1434112, 34895772, 973450000, 30636233936, 1074020373504, 41510792057176, 1753764940408768, 80412829785000000, 3977094146761424896, 211058327532167398928, 11963018212810373415168, 721321146876339731628352

Offset: 1

Author

Christian G. Bower, Jan 14 2004

Keywords

• nonn

Comments

As Alois P. Heinz has pointed out, the e.g.f in the Example section does not match the offset. However, the identity a(n) = A091481(n)/n holds with the present offset of 1. - N. J. A. Sloane, Jun 23 2017

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 290*x^4/4! + 3996*x^5/5! +...
		

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).

Links

• Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
• Index entries for sequences related to cacti

Crossrefs

Cf. A091481, A091487, A000085.

Programs

Mathematica

CoefficientList[1/x InverseSeries[x/Exp[x+x^2/2]+O[x]^20], x] Range[0, 18]! (* Jean-François Alcover, Aug 06 2018 *)

Formula

a(n) = A091481(n)/n.

From Paul D. Hanna, Jun 01 2012: (Start)

E.g.f.: (1/x)*Series_Reversion( x/exp(x+x^2/2) ).

E.g.f. satisfies: A(x) = exp( x*A(x) + x^2*A(x)^2/2 ).

E.g.f. satisfies: A( x/exp(x+x^2/2) ) = exp(x+x^2/2).

(End)

a(n+1) = n! * Sum_{k=0..n} (1/2)^(n-k) * (n+1)^(k-1) * binomial(k,n-k)/k!. - Seiichi Manyama, Aug 19 2023